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3.5. The Implementation of the Finite Element Method: Part 2 153

For P = P 2 (K) and d = 2 we get, by the shape functions λ i (2λ i − 1), the
weights 0 for the nodes a i and, by the shape functions 4λ i λ j ,the weights
1
ω i = vol (K)for b i = a ij ,i, j =1,... , 3 ,i > j ,
3
so that we have obtained here a quadrature formula that is superior to
(3.118) (for d =2). However, for d ≥ 3 this ansatz leads to negative weights
and is thus useless. We can also get the exactness in P 1 (K) by a single
quadrature node, by the barycentre (see (3.52)):

d+1
1
ω 1 =vol (K)and b 1 = a S = a i ,
d +1
i=1
which is obvious due to (3.117).
As a formula that is exact for P 2 (K) and d =3 (see [53]) we present
R =4, ω i = 1 vol (K), and the b i are obtained by cyclic exchange of the
4
barycentric coordinates:

√ √ √ √
5 − 5 5 − 5 5 − 5 5+3 5
, , , .
20 20 20 20
ˆ
On the unit cuboid K we obtain nodal quadrature formulas, which are
ˆ
exact for Q k (K), from the Newton–Cˆotes formulas in the one-dimensional
situation by

= for ˆ = i 1 ,... , i d (3.119)
ˆ ω i 1 ...i d ˆ ω i 1 ··· ˆω i d b i 1 ...i d
k k
for i j ∈{0,...,k} and j =1,... ,d.
1
are the weights of the Newton–Cˆotes formula for f(x)dx
0
Here the ˆω i j
(see [30, p. 128]). As in (3.118), for k = 1 we have here a generalization
of the trapezoidal rule (cf. (2.38), (8.31)) with the weights 2 −d in the 2 d
vertices. From k = 8 on, negative weights arise. This can be avoided and
the accuracy for a given number of points increased if the Newton–Cˆotes
integration is replaced by the Gauss–(Legendre) integration: In (3.119), i j /k
has to be replaced by the jth node of the kth Gauss–Legendre formula
.In thisway, by
(see [30, p. 156] there on [−1, 1]) and analogously ˆω i j
ˆ
ˆ
d
(k +1) quadrature nodes the exactness in Q 2k+1 (K), not only in Q k (K),
is obtained.
Now the question as to which quadrature formula should be chosen arises.
For this, different criteria can be considered (see also (8.29)). Here, we re-
quire that the convergence rate result that was proved in Theorem 3.29
should not be deteriorated. In order to investigate this question we have
to clarify which problem is solved by the approximation ¯u h ∈ V h based on
quadrature. To simplify the notation, from now on we do not consider
boundary integrals, that is, only Dirichlet and homogeneous Neumann